`cos^(-1)x = tan^(-1)x`, then: a. `x^2=((sqrt(5)-1)/2)` b. `x^2=((sqrt(5)+1)/2)` c. `sin(cos^(-1)x)=((sqrt(5)-1)/2)` d. `tan(cos^(-1)x)=((sqrt(5)-1)/2)`

int(1)/(1+cos^(2)x)dx=(1)/(sqrt(2))tan^(-1)((tan x)/(sqrt(2)))+c

cos^(-1)x=2sin^(-1)sqrt((1-x)/(2))=2cos^(-1)sqrt((1+x)/(2))

prove that cos^(-1)x=2sin^(-1)sqrt((1-x)/(2))=2cos^(-1)sqrt((1+x)/(2))

y=sin^(-1)((x)/(sqrt(1+x^(2))))+cos^(-1)((1)/(sqrt(1+x^(2))))

tan^(-1)(sqrt((a-x)/(a+x)))=(1)/(2)cos^(-1)((x)/(a))

tan (sec ^ (- 1) x) = sin (cos ^ (- 1) ((1) / (sqrt (5))))

IfI=int(dx)/(x^(3)sqrt(x^(2)-1)), then Iequals a.(1)/(2)((sqrt(x^(2)-1))/(x^(3))+tan^(-1)sqrt(x^(2)-1))+C b.(1)/(2)((sqrt(x^(2)-1))/(x^(2))+x tan^(-1)sqrt(x^(2)-1))+Cc(1)/(2)((sqrt(x^(2)-1))/(x^(2))+tan^(-1)sqrt(x^(2)-1))+Cd(1)/(2)((sqrt(x^(2)-1))/(x^(2))+tan^(-1)sqrt(x^(2)-1))+C

(tan^(-1)x)/(sqrt(1-x^(2))) withrespectto sin ^(-1)(2x sqrt(1-x^(2)))

If int(dx)/(cos^(3)x sqrt(sin2x))=a(tan^(2)x+b)sqrt(tan x)+c, then (a) a=(sqrt(2))/(5),b=(1)/(sqrt(5))(b)a=(sqrt(2))/(5),b=5(c)a=(sqrt(2))/(5),b=-(1)/(sqrt(5))(d)a=(sqrt(2))/(5),b=sqrt(5)

If I=int(sin x+sin^(3)x)/(cos2x)dx=P cos x+Q log|f(x)|+R then (a)P=(1)/(2),Q=-(3)/(4sqrt(2))(b)P=(1)/(4),Q=(1)/(sqrt(2)) (c) f(x)=(sqrt(2)cos x+1)/(sqrt(2)cos x-1)(d)f(x)=(sqrt(2)cos x-1)/(sqrt(2)cos x+1)